How to prevent unused cameras from playing their animations?
A camera itself doesn't play an animation, only the active camera determines what is visible in the rendered animation. 

You could ignore the other (inactive) camera it wouldn't be visible in the rendering if it disturbs in the viewport. 

Besides simply switching it to invisible in the outliner (as you already described) you could move each camera with its path to another layer. M layer# (0-9) 

Assuming all scene data is on layer 1 

Layer 1 Scene objects
Layer 2 Camera1 and Path1
Layer 3 Camera2 and Path2




Another possibility would be create different scenes but this would be harder to maintain.